If you want the best shrimp burger hands down you need to drive over to the shrimp shack on Sea Island Parkway. Worth the drive believe me!
While you are there you can check out Hunting Island State Park and Beach.
Shrimp Shack is a must for locals, I have been going there ever since I was a small child! They are seasonal though, but our seasons begin a lot earlier than most. The building is a blue "shack" that you can sit in at picnic tables and eat or eat outside. It is very beachy and fun! BlueFariy is right it is not on hilton head though, it is quite a drive.
BUT if you are going to visit Beaufort, Plums ( laid back and family friendly) is great. They are just finishing up a huge renovation and if you are looking for seafood, Dockside in Port Royal is where most people go. It's amazing being a seaside community we dont have more seafood places. If you visit downtown Beaufort don't forget to visit The Chocolate Tree- If you or anyone in your family is a chocolate lover you wont be sorry!
San Miguels on HH has great mexican as well. Coligny plaza (by aunt chiladas) has some good little places to eat as well. Have a great trip!